Monday, October 19, 2009

The famous Mandelbrot set is defined by the recurrent relation:
$$z_{n+1}=z_n^2+c.$$
Despite at the first glance this formula may seem simple, the set of points c, defining stable orbits of z, has incredibly complex structure:

However, when the first impression of infinite variation fades, you'll probably notice that the whole image is based on the same pattern, repeated with several variations (well, that's why it is called fractal, after all). And it is pretty boring.
Equations, other than canonical \(z_{n+1}=z_n^2+c\), can produce some different patterns,

Cubic Mandelbrot variation.

But the difference is not really drastic.
Non-homomorphic mapping functions produce significantly different patterns, but they are often too "dirty":

A Mandelbrot-like fractal, obtained from the non-homomorphic function.

This can be explained in the following way: on each iteration, non-homomorphic function scales small vicinity of each point non-uniformly. And even if the non-uniformity is small, after a number of iterations elements become "squeezed" in one directions, producing the "smeared" effect.

So, where the novelty hides?

Meet the idea: the irregular Mandelbrot set.
Instead of repetitive iteration of the same function, iterate two (or more), in an never-repeating pattern:
$$z_{n+1}=f_{Idx(n)}(z_n)+c.$$
Where \(Idx(n)\) is a non-periodic function, returning integers in the range [1..k], and \(f_1(z),...,f_k(z)\) are the different mapping functions. Non-periodicity guarantees that there are no stable cycles in Z.
Results seem interesting, at least for me. Because the mapping functions are homomorphic, there is no effect of "smearing", but non-periodicity destroys all of the repetitive, stable patterns. Here are few sample pictures:

Black-and-white palette is not a requirement, but to my artistic sense, these fractals look better without colors. See more in the album.